// Numbas version: exam_results_page_options
{"name": "Ida Friestad's copy of Integration: Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "advice": "<p>This exercise is best solved by using substitution.</p>\n<p>Note that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$</p>\n<p>So if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$</p>\n<p>Hence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$</p>\n<p>Hence the integral becomes:</p>\n<p>\\[\\begin{eqnarray*} I&amp;=&amp;\\int\\;\\frac{du}{u}\\\\ &amp;=&amp;\\ln(|u|)+C\\\\ &amp;=&amp; \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]</p>\n<p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]</p>", "name": "Ida Friestad's copy of Integration: Indefinite integral by substitution", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n    <p>Find the following integral.</p>\n  <p>Input the constant of integration as $C$.</p>\n  <p>Input all numbers as integers or fractions.</p>\n  <p>&nbsp;</p>\n    ", "parts": [{"gaps": [{"vsetrange": [0, 1], "notallowed": {"strings": ["."], "showStrings": false, "message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "partialCredit": 0}, "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showFeedbackIcon": true, "type": "jme", "vsetrangepoints": 5, "marks": 3, "showpreview": true, "checkvariablenames": false, "showCorrectAnswer": true, "answer": "ln(abs({a}*x^2+{b}*x+{c}))+C", "variableReplacements": [], "checkingtype": "absdiff", "answersimplification": "std", "scripts": {}}], "steps": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "type": "information", "scripts": {}, "prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$</p>"}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": 1, "variableReplacements": [], "showFeedbackIcon": true, "type": "gapfill", "scripts": {}, "prompt": "\n        <p>\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]</p>\n      <p>$I=\\;$[[0]]</p>\n      <p>Input the constant of integration as $C$.</p>\n      <p>Input all numbers as integers or fractions not as decimals.</p>\n      <p>Click on Show steps if you need help. You will lose 1 mark if you do so.</p>\n        "}], "variables": {"a": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1..5)", "name": "a"}, "b1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "s1*random(1..5)", "name": "b1"}, "s1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1"}, "f": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "-a*(1+b1)^2", "name": "f"}, "b": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "2*a+b1", "name": "b"}, "c1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "max(-10,f+1)+random(1..5)", "name": "c1"}, "c": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "a*b1^2+c1", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "f", "s1", "b1", "c1"], "metadata": {"description": "<p>Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}]}]}], "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}]}